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Benchmark of GW approaches for the GW100

testset

Fabio Caruso,† Matthias Dauth,‡ Michiel J. van Setten,¶ and Patrick Rinke∗,§

Department of Materials, University of Oxford, Parks Road,Oxford OX1 3PH, United Kingdom,

Theoretical Physics IV, University of Bayreuth, D-95440 Bayreuth, Germany, Nanoscopic

Physics, Institute of Condensed Matter and Nanosciences, Université Catholique de Louvain,

1348 Louvain-la-Neuve, Belgium, and COMP/Department of Applied Physics, Aalto University,

P.O. Box 11100, Aalto FI-00076, Finland

E-mail: patrick.rinke@aalto.fi

Abstract

For the recentGW100 test set of molecular ionization energies, we present a compre-

hensive assessment of differentGW methodologies: fully self-consistentGW (scGW), quasi-

particle self-consistentGW (qsGW), partially self-consistentGW0 (scGW0), perturbativeGW

(G0W0) and optimizedG0W0 based on the minimization of the deviation from the straight-line

error (DSLE-minimizedGW). We compare ourGW calculations to coupled-cluster singles,

doubles, and perturbative triples [CCSD(T)] reference data for GW100. We find scGW and

qsGW ionization energies in excellent agreement with CCSD(T), with discrepancies typically

smaller than 0.3 eV (scGW) respectively 0.2 eV (qsGW). For scGW0 andG0W0 the deviation

∗To whom correspondence should be addressed†Department of Materials, University of Oxford, Parks Road,Oxford OX1 3PH, United Kingdom‡Theoretical Physics IV, University of Bayreuth, D-95440 Bayreuth, Germany¶Nanoscopic Physics, Institute of Condensed Matter and Nanosciences, Université Catholique de Louvain, 1348

Louvain-la-Neuve, Belgium§COMP/Department of Applied Physics, Aalto University, P.O. Box 11100, Aalto FI-00076, Finland

1

from CCSD(T) is strongly dependent on the starting point. Wefurther relate the discrepancy

between theGW ionization energies and CCSD(T) to the deviation from straight line error

(DSLE). In DSLE-minimizedGW calculations, the DSLE is significantly reduced, yielding a

systematic improvement in the description of the ionization energies.

1 Introduction

Many-body perturbation theory provides an ideal frameworkfor the first-principles study of elec-

tronic excitations in molecules and solids.1 At variance with approaches based on density-functional

theory (DFT),2,3 the description of electronic many-body interactions through the electron self-

energy facilitates a seamless account of exact exchange andscreening, which are essential to

predict electronic excitations with quantitative accuracy.4–7 The GW approximation8,9 provides

an ideal compromise between accuracy and computational cost and it has, thus, evolved into the

state-of-the-art technique for the computation of ionization energies and band gaps in molecules

and solids.5

GW calculations are typically based on first-order perturbation theory (G0W0),9 a procedure

that introduces a spurious dependence of the results on the starting point, that is, the initial ref-

erence ground state the perturbation is applied to.6,10–13 The starting-point dependence may be

reduced by resorting to partial self-consistent approaches,11,14 such as eigenvalue self-consistent

GW or self-consistentGW0 (scGW0), and it is completely eliminated in the self-consistentGW

method (scGW)15,16– in which the Dyson equation is solved fully iteratively – and in quasi-particle

self-consistentGW (qsGW).17–19 While scGW implementations are still relatively rare,15,16,20–27

qsGW is now widely used.19,28–32Moreover, with rare exceptions,25 scGW and qsGW are typi-

cally not implemented in the same code and have therefore notbeen systematically compared.

Given the various flavors of the self-consistentGW methodology, benchmark and validation are

important instruments to (i) quantify the overall accuracyof GW calculations; (ii) reveal the effects

of different forms of self-consistency; (iii) identify newways to improve over existing techniques

for quasiparticle calculations. TheGW100 set provides an ideal test-case for addressing these

2

challenges.33 This benchmark set is specifically designed to target the assessment of ionization

energies and it is composed of 100 molecules of different bonding types, chemical compositions,

and ionization energies.

In this manuscript, we present the ionization energies for the molecules ofGW100 test set

calculated withG0W0, scGW0, scGW, and qsGW. We analyse their behaviour in terms of the

change in the electron density, the screening properties and the treatment of the kinetic energy.

The accuracy of differentGW approaches is established based on the comparison with coupled-

cluster singles, doubles, and perturbative triples34–36 [CCSD(T)] energies obtained for the same

geometries and basis sets.37 Our study reveals that scGW and qsGW ionization energies differ on

average by 0.3 eV and 0.15 eV from the CCSD(T) reference data,respectively. The discrepancy

of G0W0 and scGW0 from CCSD(T), on the other hand, is contingent on the starting point. For the

GW100 set, we report an average starting-point dependence of 1and 0.4 eV forG0W0 and scGW0,

respectively. Correspondingly, the starting point introduces an additional degree of freedom that

allows one to improve the agreement with CCSD(T), e.g., by imposing the satisfaction of exact

physical constraints. One of such constraint is the linearity of the total energy at fractional particle

numbers.38 The deviation from straight line error (DSLE) has been shownto lead to systematic

errors in DFT, such as the tendency to overly localize or delocalize the electron density.39,40Within

the context ofGW calculations, the DSLE may be minimized by varying the starting point. This

procedure we refer to as the DSLE-minimizedGW approach (DSLE-min).41 We show here that

DSLE-minGW reduces the discrepancy with CCSD(T) for theGW100 set as compared to scGW

with an average absolute deviation slightly larger than that of qsGW (0.26 eV, based on the def2-

TZVPP basis set). Overall, our results provide a comprehensive assessment of the starting-point

dependence, the accuracy ofG0W0 and self-consistentGW methods, and suggest that the DSLE

minimization may provide a strategy to improve the accuracyof the GW method at the cost of

G0W0 calculations.

The manuscript is organized as follows. In Sec. 2 we review the basics of theGW method

and self-consistency. The ionization energies for theGW100 test set are reported in Sec. 4, and

3

discussed in Sec. 5. DSLE-minGW results are discussed in Sec. 6. Conclusions and final remarks

are presented in Sec. 7.

2 Methods

In the following, we give a brief introduction to theGW methodology employed throughout the

manuscript: scGW, scGW0, qsGW, perturbativeG0W0, and DSLE-minGW.

In the scGW approach, the interacting Green’s functionG is determined through the iterative

solution of Dyson’s equation

G−1 = G−10 − [Σ−v0+∆vH] . (1)

∆vH denotes the change of the Hartree potential, which accountsfor the density difference between

G0 andG, andv0 is the exchange-correlation potential of the preliminary calculation. The non-

interacting Green’s functionG0 may be expressed as

Gσ0 (r,r

′,ω) = ∑

n

ψnσ (r)ψ∗nσ (r

′)

ω − (εnσ −µ)− iη sgn(µ − εnσ )(2)

whereµ is the Fermi energy, andη a positive infinitesimal.ψnσ andεnσ denote a set of single-

particle orbitals and eigenvalues determined from an independent-particle calculation (e.g., Hartree-

Fock, or DFT) for spin-channelσ . In theGW approximation, the self-energyΣ is given by

Σσ (r,r′,ω) = i∫

dω ′

2πGσ (r,r′,ω +ω ′)W(r,r′,ω ′)eiωη

. (3)

The screened Coulomb interactionW, in turn, is also determined from the solution of a Dyson-like

4

equation

W(r,r′,ω) = v(r,r′)+∫

dr1dr2v(r,r1)χ0(r1,r2,ω)W(r2,r′,ω) , (4)

wherev(r,r′) = |r− r′|−1 is the bare Coulomb interaction. The polarizabilityχ0 is most easily

expressed on the time axisτ

χ0(r,r′,τ) =−i ∑σ

Gσ (r,r′,τ)Gσ (r′,r,−τ) . (5)

and is Fourier transformed to the frequency axis before it isused in Eq. (??).

The structure of Eqs. (??)-(??) reveals the self-consistent nature of theGW approximation. Due

to the interdependence ofG, χ0, W, andΣ, Eqs. (??)-(??) need to be solved iteratively until the

satisfaction of a given convergence criterion.16 We denote the procedure in which Eqs. (??)-(??)

are solved fully self-consistently as scGW. Recent studies have revealed that Hedin’s equations

may exhibit multiple-solution behaviour.42–47For closed shell molecules we have not yet observed

multiple solutions. Moreover, it has been shown that, if self-consistency is achieved through the

solution of the Dyson equation, as in this work, the self-consistent loop converges to the unique

physical solution.46

In scGW0 the screened interactionW is evaluated only once using orbitals and eigenvalues from

an independent-particle calculation. The Dyson equation is thus solved iteratively updatingG and

Σ at each step, but keepingW0 fixed. In scGW and scGW0, the physical properties of the system –

such as, e.g., the total energy,20,21,48–50the electron density,51 and the ionization energy15,23– may

be extracted directly from the self-consistent Green’s function by means of the spectral function

A(ω) =−1π

∫

dr limr′→r

ImG(r,r′,ω) . (6)

As an example, we report in Fig. 1 the spectral function of theadenine nucleobase (C5H5N5O)

5

Figure 1: Spectral function of the adenine nucleobase, for which the molecular geometry in shown,obtained from scGW, scGW0@HF, and scGW0@PBE using the def2-TZVPP basis set.52 Thequasiparticle HOMO is indicated by arrows.

evaluated using scGW, scGW0@HF, and scGW0@PBE. For each approach, the energy of the quasi-

particle HOMO is given by the position of the highest energy peak, indicated by arrows in Fig. 1.

We note that scGW0 still exhibits a dependence on the starting point, which stems from the non-

self-consistent treatment ofW, whereas scGW is completely independent of the initial reference

calculation.15

In theG0W0 approach, the quasiparticle energiesεQP are evaluated as a first-order perturbative

correction to a set of single-particle (SP) eigenvaluesεSP [obtained, for instance, from DFT]

εQPnσ = εSP

nσ + 〈ψσn |Σ(ε

QPnσ )−v0|ψσ

n 〉 . (7)

Owing to the perturbative nature of Eq. (??), one would expect a pronounced dependence of

εQPnσ on the starting point, that is, on the set of eigenvaluesεSP

nσ and orbitalsψnσ . To bench-

mark the starting point dependence for theGW100 test set we consider hereafter two different

starting points: Hartree-Fock and the Perdew-Burke-Ernzerhof53 (PBE) generalized gradient ap-

proximation to DFT. We explicitly denote the starting-point dependence by adopting the notation

method@starting point(e.g.,G0W0@PBE).

In the qsGW self-consistency treatment the Green’s function keeps theanalytic structure of a

6

non-interacting Green’s function (omitting spin indices for brevity)

GqsGW0 (r,r′,ω) = ∑

n

ψQPn (r)ψ∗QP

n (r′)

ω − (εQPn −µ)− iη sgn(µ − εQP

n ). (8)

The quasi-particle orbitals and energies are iteratively updated solving the quasi-particle equation

applying a linear mixing scheme.17–19 The QP-orbitals of the(i + 1)th iterationψ(i+1)n (r) are

expressed in terms of the orbitals of the previous iteration

ψ(i+1)n (r) = ∑

nA

(i+1)nn ψ(i)

n (r). (9)

In the reference basisψ(i)n (r) Eq. (??) takes the form of an eigenvalue problem

∑n

A(i+1)

n′n

[

∫

dr dr′ ψ(i)n (r)

(

H0[G(i)0 ]δ (r− r′)

+Σ(r,r′))

ψ(i)n (r′)

]

= εQP(i+1)n′ A

(i+1)n′n (10)

whereH0[GqsGW0 ] is the single-particle part of the Hamiltonian evaluated with the electron density

generated byGqsGW0 . The self-energy matrix is approximated as static and Hermitian

Σnn′ =12(Σnn′(εn)+Σnn′(εn′)) . (11)

The diagonalization of Eq. (10) updatesεQP(i+1)n′ andA

(i+1)n′n . With these new orbitals, the wave

functions at iterationi + 1 (ψQP(i+1)n (r)) are constructed via Eq. (??). The orbitals become or-

thonormal by construction due to the hermiticity of the operators in Eq. (10).

qsGW is closely related toG0W0 in the sense that in each cycle of the self-consistent solution

the Green’s function is a non-interactingG0. The final result was shown to be independent of

the starting point,19 but both the stability of the iterative cycle and the rate of convergence can

be greatly improved by using an optimal starting point. In addition, it was found that a simple

iteration scheme may not always converge. In practice linear mixing scheme is applied. In qsGW

7

the orbital energies are directly available via Eq. (10).

Beside scGW, scGW0, and qsGW, other approximate self-consistentGW approaches have been

investigated in the past, such as eigenvalue self-consistent GW,11,14,54,55andGW+COHSEX.13,56

These will not be discussed in this article.

Among the different flavors ofGW calculations, the starting-point dependence is most pro-

nounced inG0W0, since bothG0 andW0 depend explicitly on the initial set of orbitals and eigen-

values. Yet, this ambiguity also provides a means to improvethe accuracy ofG0W0, by seeking

the optimal starting point that leads to the satisfaction ofexact physical constraints. A prominent

example is the piecewise linearity of the total energy.57 Usually approximate theories do not au-

tomatically exhibit a linearly changing total energy underfractional electron removal (or addition)

but instead produce a DSLE. If the total energy were a linear function of the fractional particle

number, the ionization energy of the neutral system would beequal to the electron affinity of the

cation (EAc).40,58 Identifying the ionization energy with theG0W0 quasiparticle HOMO and EAc

with the LUMO of the cationic system, one may thus define the DSLE as41

∆DSLE≡ εQPHOMO− εQP

LUMO,c. (12)

This definition can be applied to approximately quantify theDSLE in theGW method without

explicitly invoking the total energy at fractional particle numbers. Furthermore, the minimization

of ∆DSLE in G0W0 calculations allows one to find a starting point that minimizes or completely

eliminates the DSLE. We here adopt the DSLE-minGW approach proposed in Ref. 41 which is

based on these concepts. For the DSLE-min procedure we utilize starting points from PBE-based

hybrid (PBEh) functionals59 with an adjustable fractionα of Hartree-Fock exchange and evaluate

Eq. (??) with the G0W0@PBEh(α) quasiparticle energies. We then identify the optimal starting

point with the veryα that leads to a minimization of∆DSLE.

The coupled cluster singles, doubles, and perturbative triples [CCSD(T)]34–36approach is of-

ten regarded as thegold standardamong the quantum chemistry methods as it yields results that

8

approach chemical accuracy for a variety of physical/chemical properties, such as binding energies

and atomization energies. CCSD(T) values are thus particularly suitable to unambiguously estab-

lish the accuracy ofGW approaches for the ionization energies. In the following, our calculated

ionization energies are compared to reference values from CCSD(T),37 whereby the ionization en-

ergy has been obtained as a total energy difference between the ionized and neutral molecules. The

comparison to CCSD(T) is here preferred to experimental data as it allows us to focus on the effects

of exchange and correlation. We can therefore safely ignorethe effects of temperature, nuclear vi-

brations, and interaction with the environment, which affect experimental ionization energies.60

The CCSD(T) calculations of Ref. 37 used the molecular geometries of theGW100 test set,33 and

are therefore suitable to be compared with our calculations, in which the same geometries were

employed. Additional details on the CCSD(T) calculations may be found in Ref. 37.

3 Computational details

Our G0W0, DSLE-min GW, and scGW calculations have been performed with theFHI-aims

code,61–63whereas qsGW calculations have been performed using a local version of the TURBO-

MOLE64 code. ForG0W0, DSLE-min GW, and scGW the frequency dependence is treated on

the imaginary frequency axis and the quasiparticle energies are extracted by performing an ana-

lytic continuation based on Padé approximants. Similarly to Ref. 33, forG0W0 and DSLE-min

GW the parametrization of the analytic continuation employed200 imaginary frequency points on

a Gauss-Legendre grid and 16 poles for the Padé approximant method. The qsGW calculations

were performed directly in real frequency by exploiting thefull analytic structure ofG andW as

described in Ref. 19,65. Our scGW calculations used the same computational parameters as Ref.

15,16 for the frequency dependence. At variance with Ref. 33, no basis set extrapolation scheme

has been employed in this work. Additional details on the numerical implementations ofG0W0,

scGW, and scGW0 in FHI-aims15,16,63and the qsGW implementation in TURBOMOLE19,65

can be found elsewhere. All calculations use the same parameters reported in Ref. 33 for the

9

resolution-of-identity, and the real-space grids. To enable the direct comparison with reference

values from CCSD(T), we used the Gaussian def2-TZVPP basis sets.52 In FHI-aims the Gaussian

basis function are numerically tabulated and are treated asnumerical orbitals. We refer to Ref. 33

for detailed convergence tests for this procedure. For the DSLE-min method, basis set converged

calculations for the quasiparticle energies have been performed using the Tier 4 basis sets aug-

mented by Gaussian aug-cc-pV5Z basis functions (Tier 4+).63 To facilitate the comparison with

CCSD(T), we also report DSLE-min quasiparticle energies obtained with def2-TZVPP basis sets.

We use the same geometries as in Ref. 33.1 We assume zero electronic temperature and the

effects of nuclear vibrations are ignored. All ionization energies are vertical and do not include

any relativistic corrections.

4 Ionization energies for the GW100 set

TheGW100 test set consists of 100 atoms and molecules which have been selected to span a broad

range of chemical bonding situations, chemical compositions, and ionization energies. Due to the

absence of all-electron def2-TZVPP basis sets for fifth-rowelements, we exclude Xe, Rb2, Ag2,

and the iodine-containing compounds (I2, C2H3I, CI4, and AlI3). For the remaining 93 member of

GW100 we can then conduct a meaningful comparison with CCSD(T)reference data.

As discussed in Ref. 33, many molecules of the GW100 testset have positive LUMO energies

(that is, negative electron affinities), which makes them unsuitable for a systematic assessment of

electron affinities since experimental data for such compounds is difficult to obtain. Moreover,

CCSD(T) reference data is presently also not available for the LUMOs in the GW100 testset.37

For these reasons, we focus here on the first vertical ionization energy, for which experimental and

CCSD(T) reference data are available. An assessment of GW methods for electron affinities may

found in Ref. 13. In table 1, we report the ionization energies for this subset ofGW100 calculated

1Experimental geometries have been employed whenever available, otherwise molecular geometries are optimizedwithin the PBE approximation for the exchange-correlationfunctional using the def2-QZVP basis set. More detailson the strategy adopted for selecting the compounds of theGW100 set and their geometries are given in Ref. 33.

10

with qsGW, scGW, scGW0@HF, and scGW0@PBE and def2-TZVPP basis sets. For comparison,

we also report the CCSD(T) ionization energies from Ref. 37.

Table 1: Vertical ionization energies for theGW100 test set calculated with qsGW, scGW,scGW0@HF, scGW0@PBE, and DSLE-minimizedG0W0 (DSLE-min) and def2-TZVPP basis sets.Basis set converged DSLE-minimizedG0W0 calculations employ Tier 4 basis sets augmented byGaussian aug-cc-pV5Z basis functions, denoted as DSLE-min(T4+). For comparison, we alsoreport CCSD(T) values from Ref.37 All values are in eV.

Name Formula qsGW scGW scGW0@HF scGW0@PBE DSLE-min DSLE-min (T4+) CCSD(T)

1 Helium He -24.43 -24.44 -24.47 -24.01 - - -24.51

2 Neon Ne -21.62 -21.40 -21.49 -20.84 -20.82 -20.22 -21.32

3 Argon Ar -15.53 -15.26 -15.50 -15.18 -15.34 -15.27 -15.54

4 Krypton Kr -13.74 -13.65 -13.88 -13.62 -13.62 -13.75 -13.94

6 Hydrogen H2 -16.22 -16.18 -16.27 -15.98 - - -16.40

7 Lithium dimer Li2 -5.34 -4.96 -5.15 -5.02 -5.00 -5.05 -5.27

8 Sodium dimer Na2 -5.02 -4.63 -4.80 -4.74 -4.87 -4.93 -4.95

9 Sodium tetramer Na4 -4.25 -3.85 -4.09 -3.99 -4.18 -4.27 -4.23

10 Sodium hexamer Na6 -4.41 -3.94 -4.25 -4.16 -4.31 -4.40 -4.35

11 Dipotassium K2 -4.08 -3.73 -3.90 -3.86 -3.96 -4.10 -4.06

13 Nitrogen N2 -16.01 -15.44 -15.84 -15.32 -15.49 -15.75 -15.57

14 Phosphorus dimer P2 -10.40 -9.73 -10.20 -10.01 -10.30 -10.52 -10.47

15 Arsenic dimer As2 -9.62 -9.00 -9.48 -9.34 -9.52 -9.82 -9.78

16 Fluorine F2 -16.33 -15.78 -16.17 -15.50 -15.56 -15.76 -15.71

17 Chlorine Cl2 -11.52 -11.07 -11.47 -11.13 -11.36 -11.55 -11.41

18 Bromine Br2 -10.54 -10.23 -10.58 -10.30 -10.31 -10.77 -10.54

20 Methane CH4 -14.56 -14.28 -14.50 -14.14 -14.20 -14.35 -14.37

21 Ethane C2H6 -12.99 -12.62 -12.92 -12.55 -12.60 -12.75 -13.04

22 Propane C3H8 -12.35 -11.95 -12.30 -11.92 -12.03 -12.18 -12.05

23 Butane C4H10 -11.89 -11.46 -11.85 -11.46 -11.73 -11.88 -11.57

24 Ethylene C2H4 -10.63 -10.14 -10.45 -10.24 -10.40 -10.60 -10.67

25 Acetylene C2H2 -11.53 -10.89 -11.23 -10.98 -11.17 -11.43 -11.42

26 tetracarbon C4 -11.45 -10.68 -11.21 -10.87 -10.87 -11.07 -11.26

27 Cyclopropane C3H6 -11.13 -10.62 -10.98 -10.66 -10.77 -11.00 -10.87

28 Benzene C6H6 -9.38 -8.73 -9.20 -8.97 -9.12 -9.34 -9.29

29 Cyclooctatetraene C8H8 -9.30 -7.81 -8.33 -8.04 -8.21 -8.44 -8.35

30 Cyclopentadiene C5H6 -8.73 -8.10 -8.54 -8.29 -8.47 -8.69 -8.68

31 Vynil fluoride C2H3F -10.64 -10.11 -10.46 -10.16 -10.36 -10.59 -10.55

32 Vynil chloride C2H3Cl -10.09 -9.63 -10.02 -9.72 -9.92 -10.14 -10.09

33 Vynil bromide C2H3Br -9.33 -8.82 -9.19 -8.94 -9.06 -9.32 -9.27

35 Carbon tetrafluoride CF4 -16.77 -16.34 -16.75 -15.89 -15.84 -15.78 -16.30

36 Carbon tetrachloride CCl4 -11.63 -11.16 -11.69 -11.18 -11.46 -11.57 -11.56

37 Carbon tetrabromide CBr4 -10.57 -10.10 -10.59 -10.16 -10.33 -10.59 -10.46

39 Silane SiH4 -13.04 -12.74 -13.00 -12.55 -12.66 -12.88 -12.80

40 Germane GeH4 -12.81 -12.40 -12.67 -12.28 -12.41 -12.55 -12.50

41 Disilane Si2H6 -10.88 -10.46 -10.82 -10.45 -10.48 -10.75 -10.65

42 Pentasilane Si5H12 -9.56 -9.04 -9.50 -9.10 -9.18 -9.32 -9.27

43 Lithium hydride LiH -8.00 -7.88 -7.97 -7.45 -6.48 -6.71 -7.96

44 Potassium hydride KH -6.17 -6.02 -6.17 -5.52 -5.65 -5.63 -6.13

45 Borane BH3 -13.52 -13.22 -13.42 -13.05 -13.17 -13.30 -13.28

46 Diborane(6) B2H6 -12.58 -12.23 -12.54 -12.09 -12.17 -12.30 -12.26

47 Ammonia NH3 -11.08 -10.76 -10.97 -10.59 -10.59 -10.78 -10.81

Continues on next page

11

Table 1 continued

Name Formula qsGW scGW scGW0@HF scGW0@PBE DSLE-min DSLE-min (T4+) CCSD(T)

48 Hydrogen azide HN3 -10.91 -10.24 -10.69 -10.38 -10.61 -10.89 -10.68

49 Phosphine PH3 -10.65 -10.24 -10.53 -10.28 -10.39 -10.60 -10.52

50 Arsine AsH3 -10.50 -10.10 -10.39 -10.17 -10.24 -10.49 -10.40

51 Hydrogen sulfide SH2 -10.39 -9.97 -10.26 -10.02 -10.15 -10.38 -10.31

52 Hydrogen fluoride FH -16.33 -16.11 -16.26 -15.71 -15.63 -15.64 -16.03

53 Hydrogen chloride ClH -12.65 -12.28 -12.55 -12.27 -12.41 -12.57 -12.59

54 Lithium fluoride LiF -11.52 -11.34 -11.50 -10.59 -10.66 -10.85 -11.32

55 Magnesium fluoride F2Mg -13.99 -13.77 -13.97 -13.05 -12.87 -13.00 -13.71

56 Titanium fluoride TiF4 -15.75 -15.55 -16.15 -14.98 -14.80 -15.19 -15.48

57 Aluminum fluoride AlF3 -15.69 -15.40 -15.68 -14.83 -14.59 -14.75 -15.46

58 Fluoroborane BF -11.13 -10.64 -10.94 -10.56 -10.82 -10.98 -11.09

59 Sulfur tetrafluoride SF4 -12.98 -12.47 -12.95 -12.36 -12.51 -12.73 -12.59

60 Potassium bromide BrK -8.15 -7.88 -8.12 -7.72 -7.84 -8.25 -8.13

61 Gallium monochloride GaCl -9.80 -9.35 -9.69 -9.49 -9.62 -9.93 -9.77

62 Sodium chloride NaCl -9.07 -8.79 -9.03 -8.51 -8.79 -9.03 -9.03

63 Magnesium chloride MgCl2 -11.64 -11.39 -11.70 -11.24 -11.22 -11.39 -11.67

65 Boron nitride BN -11.79 -11.06 -11.58 -11.27 -11.19 -11.81 -11.89

66 Hydrogen cyanide NCH -13.65 -13.15 -13.51 -13.20 -13.47 -13.72 -13.87

67 Phosphorus mononitride PN -11.93 -11.56 -12.03 -11.60 -11.60 -11.84 -11.74

68 Hydrazine H2NNH2 -10.08 -9.63 -9.93 -9.52 -9.53 -9.75 -9.72

69 Formaldehyde H2CO -11.22 -10.82 -11.15 -10.67 -10.77 -11.02 -10.84

70 Methanol CH4O -11.46 -11.07 -11.36 -10.86 -10.94 -11.19 -11.04

71 Ethanol C2H6O -11.07 -10.69 -11.05 -10.51 -10.59 -10.84 -10.69

72 Acetaldehyde C2H4O -10.62 -10.20 -10.59 -10.03 -10.10 -10.36 -10.21

73 Ethoxy ethane C4H10O -10.23 -9.81 -10.27 -9.67 -9.77 -10.02 -9.82

74 formic acid CH2O2 -11.78 -11.42 -11.80 -11.19 -11.29 -11.57 -11.42

75 Hydrogen peroxide HOOH -11.98 -11.55 -11.90 -11.38 -11.42 -11.69 -11.59

76 Water H2O -12.91 -12.59 -12.78 -12.32 -12.26 -12.45 -12.57

77 Carbon dioxide CO2 -14.07 -13.55 -13.95 -13.45 -13.61 -13.91 -13.71

78 Carbon disulfide CS2 -10.04 -9.45 -9.95 -9.69 -9.89 -10.14 -9.98

79 Carbon oxysulfide OCS -11.33 -10.72 -11.17 -10.88 -11.08 -11.35 -11.17

80 Carbon oxyselenide OCSe -10.60 -10.00 -10.42 -10.19 -10.29 -10.62 -10.79

81 Carbon monoxide CO -14.55 -13.95 -14.43 -13.90 -14.21 -14.44 -14.21

82 Ozone O3 -13.21 -12.54 -13.16 -12.57 -12.24 -12.49 -12.55

83 Sulfur dioxide SO2 -12.54 -12.05 -12.54 -12.06 -12.21 -12.55 -13.49

84 Beryllium monoxide BeO -10.11 -9.77 -10.01 -9.58 -9.40 -9.68 -9.94

85 Magnesium monoxide MgO -8.30 -7.97 -8.27 -7.72 -7.48 -7.60 -7.49

86 Toluene C7H8 -9.00 -8.35 -8.83 -8.60 -8.74 -8.96 -8.90

87 Ethylbenzene C8H10 -8.97 -8.30 -8.80 -8.55 -8.68 -8.91 -8.85

88 Hexafluorobenzene C6F6 -9.91 -9.48 -10.08 -9.66 -9.96 -10.23 -9.93

89 Phenol C6H5OH -8.82 -8.19 -8.67 -8.39 -8.52 -8.78 -8.70

90 Aniline C6H5NH2 -8.12 -7.51 -7.99 -7.69 -7.83 -8.09 -7.99

91 Pyridine C5H5N -9.76 -9.11 -9.58 -9.37 -9.53 -9.76 -9.66

92 Guanine C5H5N5O -7.95 -7.49 -8.06 -7.71 -7.88 -8.18 -8.03

93 Adenine C5H5N5O -8.41 -7.77 -8.33 -8.00 -8.16 -8.45 -8.33

94 Cytosine C4H5N3O -8.99 -8.38 -8.93 -8.47 -8.63 -8.92 -9.51

95 Thymine C5H6N2O2 -9.30 -8.69 -9.25 -8.83 -9.01 -9.28 -9.08

96 Uracil C4H4N2O2 -9.74 -9.12 -9.66 -9.22 -9.41 -9.69 -10.13

97 Urea CH4N2O -10.45 -10.02 -10.45 -9.81 -10.13 -10.44 -10.05

99 Copper dimer Cu2 -7.52 -6.98 -7.23 -7.29 -7.15 -7.57 -7.57

100 Copper cyanide NCCu -10.97 -10.54 -11.13 -10.26 -10.38 -10.50 -10.85

12

0

20

40

60

0

20

40

Num

ber

of m

olec

ules

-1 0 1Error [eV]

0

20

40

-1 0 1Error [eV]

scGW

scGW0 @PBE

qsGW

scGW0@HF

ME = -0.30 eV

ME = 0.06 eV

ME = -0.34 eV

(a)

ME = 0.15 eV

(c)

(b)

(d)

ME = -0.69 eV

G0W0@PBE G0W0@HFME = 0.26 eV

(e) (f)

Figure 2: Error distribution [defined as the difference to CCSD(T) reference energies from Ref.37] for the ionization energies of theGW100 test set evaluated using (a) scGW, (b) qsGW, (c)scGW0@PBE, (d) scGW0@HF, (e)G0W0@PBE, and (f)G0W0@HF and def2-TZVPP basis sets.The mean error (ME) for each method is listed in the corresponding panel.

5 Comparison of GW Methods

To quantify the deviation from CCSD(T) calculations, we analyse the error∆ ≡ εHOMOCCSD(T)−εHOMO

QP

and the absolute error∆abs≡ |εHOMOCCSD(T)−εHOMO

QP |. In Fig. 2, we report the error distribution for the

molecules of theGW100 test set, whereas the absolute error is reported in Fig. 3.

5.1 scGW vs qsGW

We start by considering the scGW and qsGW approaches. At variance withG0W0 and scGW0, the

scGW ionization energies are independent of the starting point.15,16Any deviations between scGW

and CCSD(T) can then be attributed to intrinsic limitationsof theGW approximation (i.e. missing

vertex corrections) rather than the artificial starting-point dependence introduced by perturbation

13

0

20

40

60

0

20

40

Num

ber

of m

olec

ules

0 0.5 1Error [eV]

0

20

40

0 0.5 1 1.5Error [eV]

scGW

scGW0@PBE

qsGW

scGW0@HF

MAE = 0.32 eV

MAE = 0.20 eVMAE = 0.35 eV

(a)

MAE = 0.22 eV

(c)

(b)

(d)

G0W0@HFG0W0@PBE

(e) (f)

MAE = 0.69 eV MAE = 0.35 eV

Figure 3: Absolute error distribution (defined similarly toFig. 2) for the ionization energies ofthe GW100 test set evaluated using (a) scGW, (b) qsGW, (c) scGW0@PBE, (d) scGW0@HF, (e)G0W0@PBE, and (f)G0W0@HF and def2-TZVPP basis sets. The mean absolute error (MAE)foreach method is listed in the corresponding panel.

theory or approximate self-consistent procedures. The qsGW ionization energies of molecules

have also been reported to be independent of the starting point.19 However, for some solids, a

dependence on the starting point has been observed.66

Our calculations reveal that qsGW overestimates the ionization potentials in our test set by

0.15 eV on average [Fig. 2 (b)], whereas scGW underestimates them by 0.3 eV [Fig. 2 (a)]. qsGW

exhibits a MAE of∼ 0.22 eV [Fig. 3 (b)] and it thus yields quasiparticle energies in slightly better

agreement with CCSD(T) than scGW [MAE= 0.32 eV, Fig. 3 (a)]. Overall, scGW and qsGW

ionization energies differ on average by 0.45 eV, revealingthat different forms of self-consistency

may affect significantly the value of the quasiparticle energies and the corresponding agreement

with experiment. In the following we explore four differentpotential explanations.

14

Figure 4: Deviation between the CCSD(T) reference ionization energies and our first-principlescalculations obtained using (a) scGW, scGW0@PBE, and scGW0@HF, and (b) scGW, qsGW,G0W0@PBE, andG0W0@HF and def2-TZVPP basis sets. Only compounds with ionization en-ergies that differ from CCSD(T) by less than 1 eV are included. Vertical dotted lines denotes theseparation between different subgroups of theGW100 test set and coincide with the horizontalseparation lines of table 1. The separation in subgroups (aswell as the name attributed to eachsubgroup) is a guide to the eye, but not necessarily representative of the chemical compositionsof each compound. Points falling within the horizontal shaded area differ by less the 0.3 eV fromCCSD(T).

15

5.1.1 Screening properties

While it is expected that different forms of self-consistency lead to different results, the magni-

tude of the difference is surprising. At first glance, scGW and qsGW should be similar since in

both approaches the quasiparticle energies enter the denominator of the Green’s function. For

both approaches we would therefore expect underscreening,due to the inverse dependence of the

magnitude of screening on the energy difference between thelowest unoccupied and the highest

occupied state inGW. In a beyond-GW treatment this underscreening due to the large quasipar-

ticle gap would be compensated by vertex corrections, such as ladder diagrams.30,67 Without this

compensation, the underscreening due to the too large quasiparticle gap inW would lead to an

overestimation of ionization energies and quasiparticle energies that resemble those ofG0W0@HF,

which is also based on an underscreenedW0 due to the large HOMO-LUMO gap in HF. For qsGW

we indeed observe this resemblance withG0W0@HF in Fig. 2 and 3, which results in the afore-

mentioned slight average overestimation of ionization energies compared to CCSD(T). The small

reduction of the ionization energies by 0.09 eV in going fromG0W0@HF to qsGW can therefore

be attributed to a reduction of the underscreening due to thefact that the qsGW gap is smaller than

the HF gap and to density changes that we will discuss in the following.

The corresponding ionization-energy histogram for scGW is closer to scGW0@PBE andG0W0@PBE

than toG0W0@HF, with a concomitant underestimation of the CCSD(T) reference data. This

observation is consistent with previous scGW calculations for molecules11,13,15,16,25,51,68that ob-

served a similar underestimation of the ionization potential. Also in scGW the HOMO-LUMO

gaps is smaller than inG0W0@HF and smaller than in qsGW. scGW therefore underscreens less

than qsGW and we attribute part of the 0.45 eV average deviation between qsGW and scGW to

this difference in screening.

5.1.2 Spectral-weight transfer

For solids, a spectral-weight transfer from the main quasiparticle peaks to satellites has been re-

ported for scGW calculations of the homogeneous electron gas.69 Schematically, the self-consistent

16

Green’s function can be written asG= ZGqp+ G, whereZ is the spectral weight of the quasipar-

ticle peakGqp andG the incoherent part of the spectral function. In qsGW Z is equal to one andG

is zero.19,70Conversely, for scGW Zis smaller than one andG larger than zero, as spectral weight

is transferred fromGqp to G. This spectral weight transfer leads to an additional underscreening

and an overestimation of band gaps in solids.22,27,71

For small molecules there are no continuum states or collective excitations that could be ex-

cited at valence energies.49 The scGW spectral functions therefore are sharply peaked around the

quasiparticle energies and the spectrum exhibits no signature of an incoherent background in the

valence energy region49 as show in Fig. 1. We would thus not expect any additional underscreen-

ing due to spectral-weight transfer, becauseZ is equal to one andG is zero, just as for qsGW. The

spectral-weight transfer concept can therefore not explain the consistent underestimation observed

for molecules in scGW.11,13,15,16,51,68

5.1.3 Self-consistent density

Further insight into the effects of differentGW approaches on electron correlation may be gained

from the study of the self-consistent electron density. To focus on the effects of correlation, we

consider in the following differences of the PBE, scGW, scGW0, and qsGW electron density to

the density of a Hartree-Fock calculation using with the same computational parameters. Figure 5

illustrates isosurfaces of these density differences for F2 (upper panel) and BF (lower panel) with

isovalues of 0.05 and 0.01 Å−3, respectively. To quantify the difference between theGW and the

HF densities, we introduce a density difference parameterD defined as:

D =∫

dr∣

∣

∣nGW(r)−nHF(r)

∣

∣

∣(13)

for which the values for BF and F2 are also reported in Fig. 5.

For both BF and F2, scGW and scGW0 induce qualitatively similar modifications of the electron

density as compared to the Hartree-Fock reference both in shape and magnitude (as quantified by

17

Figure 5: Isosurfaces of the density difference to Hartree-Fock for PBE, scGW0@HF,scGW0@PBE, scGW, and qsGW. We used an isovalue of 0.05 and 0.01 Å−3 for F2 (upper panel)and BF (lower panel), respectively.

D). In particular, both scGW and scGW0@HF yieldD= 0.20 for BF and F2, whereas scGW0@PBE

yields a slight larger modification of the electron density,quantified from the largerD value, which

we attribute to over-screening induced by the PBE starting point. In qsGW the change of electron

density is more pronounced with respect to scGW and, for the BF dimer, exhibits a considerably

different charge redistribution pattern.

Overall, these results indicate that electron densities resulting from scGW and qsGW calcula-

tion may exhibit quantitative and qualitative differences. In self-consistent treatments, such density

difference affect the external and the Hartree potential aswell as the kinetic and the self-energy and

thus contribute to the quasiparticle energy difference observed in this work. However, the small

example shown in Fig. 5 illustrates that the density difference between qsGW and scGW is neither

systematic in shape nor in magnitude and can probably not explain the systematic shift of∼0.45

18

eV observed between our qsGW and scGW data.

5.1.4 Kinetic energy

Another aspect in which scGW and qsGW differ is the treatment of the kinetic energy. In theG0W0

approach, the quasiparticles are subject to the non-interacting kinetic energy. If the non-interacting

Green’s functionG0 derives, for example, from a Kohn-Sham DFT calculation the kinetic energy

contribution to the total energy would be that of thefictitious non-interacting system of Kohn-

Sham particles (Ts). In Kohn-Sham theory, the difference betweenTs and the kinetic energy of the

interacting systemT – as obtained for instance from a self-consistent Green’s function calculation –

is included through the exchange-correlation energy functional. In the following, we analyze how

the kinetic energy is handled in qsGW, a hybrid approach which combines elements of Green’s

theory and Kohn-Sham theory. In particular, we discuss whether the differences in the scGW and

qsGW quasiparticle energies may be ascribed to a different treatment of the kinetic energy in the

two methods.

The difference between the non-interacting and the interacting kinetic energy of aGW calcula-

tion may be quantified by invoking the analogy with the random-phase approximation (RPA).72,73

The total energy in scGW, G0W0, and RPA can be separated into different contributions:49,50

EGW[G] = T[G]+Eext[G]+EH[G]+Ex[G]+UGWc [G] (14)

EG0W0[G0] = Ts[G0]+Eext[G0]+EH[G0]+Ex[G0]+UGWc [G0] (15)

ERPA[G0] = Ts[G0]+Eext[G0]+EH[G0]+ERPAxc [G0] (16)

= Ts[G0]+Eext[G0]+EH[G0]+Ex[G0]+ERPAc [G0]+TRPA

c [G0] (17)

whereT is the fully interacting kinetic energy,Ts the non-interacting kinetic energy,Eext the exter-

nal energy,EH the Hartree energy andEx the exchange energy evaluated for the fully interacting

Green’s functionG or the non-interacting reference calculationG0. Following Ref. 49,50, we

definedUGWc andERPA

c as the correlation energy functionals in theGW and RPA approximation,

19

respectively:

UGWc =

∫ ∞

0

dω2π

Tr{v[χ(iω)−χ0(iω)]} (18)

ERPAc =

∫ ∞

0

dω2π

∫ 1

0dλ Tr{v[χλ (iω)−χ0(iω)]} (19)

whereχλ is the reducible polarizability

χλ = χ0+χ0vχλ (20)

at coupling strengthλ that follows from the irreducible polarizabilityχ0 defined in Eq. (??). In

GW there is no coupling strength integration andχ = χλ=1. RPA contains a coupling strength inte-

gration over fictitious systems with coupling strengthλ that varies between zero (non-interacting)

and one (fully interacting). The comparison between Eq. (??) and (??) reveals thatUGWc contains

only electronic correlation (that is, arising from the Coulomb interaction), whereasERPAc recap-

tures a interacting kinetic energy contribution through the coupling constant integration for the

same starting pointG0.49 We can then define the kinetic energy contribution of the correlation

energy as

TRPAc [G0]≡ ERPA

c [G0]−UGWc [G0]. (21)

Equations (??) to (??) illustrate that both the full Green’s function framework (scGW) and DFT

(e.g., RPA) incorporate the interacting kinetic energy. Inthe perturbativeG0W0 framework, how-

ever, this contribution is absent.

In scGW the quasiparticle energies are extracted directly from theimaginary part of the Green’s

function, i.e. the spectral function, as illustrated in Section 2, and therefore contain a contribution

from the interacting kinetic energy. In DFT, the Kohn-Sham eigenvalues are obtained from the

solution of the Kohn-Sham equation. The effective Kohn-Sham potential includes the exchange-

correlation potential, that is defined as the functional derivative of the exchange-correlation energy

δExcδn and therefore includes the difference between the interacting and the non-interacting kinetic

20

energy in the correlation potential via the derivative ofTc.

Conversely, in theG0W0 approach, the quasiparticle energiesεQP are evaluated as a first-order

perturbative correction to the single-particle eigenvaluesεSPas shown in Eq. (??), which we repeat

here for clarity

εQPnσ = εSP

nσ + 〈ψσn |Σ

G0W0(εQPnσ )−vxc|ψσ

n 〉 . (22)

For DFT starting points, the matrix element of the exchange-correlation potentialvxc subtracts the

aforementionedTc contribution from the eigenvalueεSPnσ . SinceΣG0W0 is purely an exchange and

Coulomb correlation self-energy, it does not add an interacting kinetic energy contribution back in,

which is thus absent from theG0W0 quasiparticle energies.

In qsGW the situation is similar toG0W0. Equation (??) is also solved for the qsGW quasipar-

ticle energies. However,vxc is replaced byΣ, the self-consistently determined, optimal, non-local,

static potential that best represents theG0W0 self-energy. SinceΣ derives fromΣG0W0 it also does

not contain an interacting kinetic energy contribution andneither doesεSPnσ . The kinetic energy

contribution is therefore also absent from the quasiparticle energies in the qsGW framework.

We therefore conclude that although scGW and qsGW at first glance appear to be similarGW

self-consistency schemes, they differ quite considerablyin their treatment of the kinetic energy. We

attribute the observed, average deviation of∼0.45 eV between these two schemes to the difference

in the kinetic energy treatment, the difference in the electron density and the screening properties.

5.2 Partially self-consistent GW

We now turn to the partially self-consistentGW0 scheme. Unlike scGW and qsGW, the ioniza-

tion energies of this partially self-consistent scheme still exhibit a dependence on the starting

point, owing to the non-self-consistent treatment ofW.11 To account for this dependence, we

based our scGW0 calculations on two different starting points: PBE and HF. Our calculations

for the GW100 set indicate that scGW0@PBE underestimate the ionization energies by 0.34 eV

[Fig. 2 (c)], whereas scGW0@HF overestimates them by 0.06 eV [Fig. 2 (d)]. This trend reflects

21

the over- and under-screening of the screened Coulomb interaction induced by the evaluation of

W with PBE or HF orbitals, respectively. In practice, owing tothe band-gap problem of Kohn-

Sham DFT74 PBE calculations typically underestimate the HOMO-LUMO gap by as much as

50% as compared to quantum-chemical calculations or reference experimental values. The small

HOMO-LUMO gap, in turn, leads to an overestimation of the polarizability [Eq. (??)] and, corre-

spondingly, of the correlation part of theG0W0 self-energy, as alluded to in the previous Section.

Conversely, HOMO-LUMO gaps are typically overestimated inHartree-Fock owing to the lack

of electronic correlation which leads, following similar arguments, to an underscreening of the

polarizability and a corresponding overestimation of the quasiparticle energies.

The scGW0@HF and scGW0@PBE ionization energies differ from each other by 0.4 eV on

average, with a maximum deviation of 1 eV (e.g., for F2Mg). scGW0@HF exhibits the lowest

MAE (0.2 eV) relative to CCSD(T) among theGW methods considered in this work [Fig. 3 (d)].

It gives larger ionization energies than scGW on average. Since also the partial self-consistency

scheme incorporates the interacting kinetic energy through the self-consistent Green’s function,

we attribute the larger ionization energies in scGW0@HF to a more pronounced underscreening

due to the fact that the HF HOMO-LUMO gap that determines the screening strength ofW@HF

is larger than that of scGW. Conversely, the underscreening in scGW0@PBE indicates that PBE-

based screening (W@PBE) is not as suitable for theGW100 set as Hartree-Fock based screening

(W@HF), although for larger molecules or solids, the situation may differ.

5.3 The perturbative G0W0 scheme

For comparison, we report in Figs. 2 and 3 the ME and MAE ofG0W0@HF andG0W0@PBE. Other

G0 starting points will be discussed in connection to the DSLE-scheme in Section 6.G0W0@PBE

underestimates the ionization energies by 0.7 eV [Fig. 2 (e)], whereasG0W0@HF overestimates

by 0.3 eV [Fig. 2 (f)]. G0W0 calculations exhibit a more pronounced dependence on the starting

point as compared to scGW0, since neitherG andW are treated self-consistently. The average

discrepancy betweenG0W0@PBE andG0W0@HF ionization energies is approximately 1 eV, and

22

can be as large as 2 eV. ForG0W0, in particular, HF provides a better starting point as it leads to a

mean absolute error a factor of 2 smaller as compared to PBE [Fig. 3 (e)-(f)]. A similar observation

was also made for the ionization energies and electron affinities of organic acceptor molecules.13

As alluded to in Section 5.1,G0W0@HF gives results that are comparable to qsGW. However,

scGW differs appreciably. Looking at the progression fromG0W0@HF to scGW0@HF to scGW we

can now understand the reduction of the ionization energiesin terms of changes to the electronic

screening, the electron density and the kinetic energy. Going from G0W0@HF to scGW0@HF

incurs a density change as illustrated in Fig. 5 and a change from the non-interacting to the inter-

acting kinetic energy (albeit without possible kinetic energy changes due to changes inW). Both

effects together reduce the ionization energies on average. Going from scGW0@HF to scGW does

not change the density appreciably anymore according to Fig. 5. The additional reduction of the

ionization energies in scGW therefore results from a reduction of the underscreening inW in going

from W@HF to the self-consistentW and a concomitant change in the kinetic energy.

5.4 Trends across the GW100 set

For all molecules of theGW100 set, the deviation from the CCSD(T) ionization energiesis il-

lustrated in Fig. 4. The horizontal shaded area marks pointsdiffering by less than 0.3 eV from

CCSD(T). As a guide through the chemical composition of the different compounds, we divided

theGW100 set into ten subgroups: atoms, dimers, hydrocarbons, hydrides, halogenides, nitrides,

oxides, aromatic molecules, nucleobases, and transition metals compounds. These categories are

intended as an approximate indication of the chemical compositions of theGW100 subsets. Dif-

ferent categories are color-coded and separated by vertical dotted lines.

Fig. 4 (b) shows that scGW provides accurate ionization energies for molecules of thehydride,

halogenide, and oxide groups. The MAE reduces to∼ 0.15 eV if we consider only molecules of

the oxide group. A common element of these compounds is the presence of highly electroneg-

ative atoms (O, F, Cl) and, correspondingly, the formation of covalent bonds with a strong ionic

character. The largest discrepancies among the scGW ionization energies are observed for systems

23

characterized predominantly by delocalizedπ-type orbitals such as, e.g., compounds of the hydro-

carbon and nucleobase groups. At variance with scGW, scGW0@HF [Fig. 4 (a)] andG0W0@HF

[Fig. 4 (b)] exhibit the largest deviation from CCSD(T) for ionic compounds (hydrides, halo-

genides, and oxides), whereas the discrepancy is small forπ-orbital compounds. Figure 4 further

reveals that scGW0@PBE deviates rather homogeneously from the CCSD(T) reference data.

6 DSLE-min GW ionization energies

We now turn to the discussion of the accuracy of basis-set converged (T4+) DSLE-minGW cal-

culations. In Fig. 6, we report the DSLE for two representative molecules of theGW100 test set,

sodium chloride (left) and the adenine nucleobase (right).In practice, the DSLE is estimated by

evaluation of Eq. (??) with the QP energies fromG0W0@PBEh(α) for α between 0 (pure PBE

exchange) and 1 (pure Hartree-Fock exchange). In addition,we show the deviation of the quasi-

particle energy for the HOMO from the CCSD(T) reference (εHOMOCCSD(T)− εHOMO

QP ). Both molecules

exhibit a clear correlation between the DSLE and the accuracy of the ionization energy. Figure 6

reveals thatα values smaller than 0.4 typically result in a positive∆DSLE and a corresponding un-

derestimation of the ionization energy, whereas the opposite trend is observed for largerα values.

At α ≈ 0.4 for NaCl andα ≈ 0.45 for adenine we find∆DSLE= 0. For NaCl, the DSLE-minimized

starting point yields an ionization energy that coincides with the CCSD(T) result, whereas for ade-

nine it is slightly overestimated. More generally, we find for all the systems in theGW100 set that

the deviation from CCSD(T) is strongly reduced when the DSLEis minimized.

More generally, we find that also for other systems of theGW100 set the deviation from

CCSD(T) is strongly reduced whenever the DSLE is minimized.In Fig. 7, we illustrate the dis-

tribution of optimalα values across the systems of the GW100 testset computed withthe Tier 4+

basis set. The optimalα determined from the DSLE-minG0W0 approach is almost unaffected

by finite basis set errors owing to cancellation effects in Eq. (??). Only three molecules of the

GW100 testset minimize the DSLE already forα = 0 (that is, for pure PBE exchange): LiH, Li2,

24

Figure 6: Correlation between DSLE and accuracy of the ionization energy for the NaCl (left)and the adenine (right) molecules. The deviation of theG0W0@PBEh(α) HOMO energies fromthe reference CCSD(T) ionization energies,εCCSD(T)− εG0W0, is displayed in blue for differentamounts of Hartree-Fock exchangeα used in the PBE hybrid starting point. The∆DSLE values aredepicted in red as a function ofα. We use Tier 4+ basis sets for our DSLE-minG0W0 calculations.

and Na2. The average over allα values amounts to 0.35. This substantiates the results of previous

starting-point benchmarks,75–78which find a similar fraction of Fock exchange to provide the most

accurate vertical ionization energies. Figure 8 explicitly shows the MAE for the ionization energies

of theGW100 set obtained fromG0W0@PBE(α) as a function ofα and, marked by a horizontal

red line, the MAE of DSLE-minGW.

Finally, in Fig. 8 we report the MAE for the ionization energies of theGW100 set obtained from

G0W0@PBE(α) as a function ofα and, marked by a horizontal red line, the MAE of DSLE-min

GW. Figure 8 reveals that, among all possible choices of PBEh(α) starting points, the DSLE-

minimization procedure yields a gratifying MAE and, thus, is a reliable choice for ionization en-

ergy predictions.

7 Conclusions

In summary, we have studied the accuracy of state-of-the-art techniques based on many-body per-

turbation theory for the description of (charged) electronic excitations in molecules. For com-

pounds of theGW100 benchmark set, we have computed the ionization energiesas obtained from

perturbative (G0W0) and self-consistentGW approaches (scGW, qsGW, and scGW0), as well from

25

0 0.2 0.4 0.6 0.8 1α

0

20

40

60

Num

ber

of m

olec

ules

Figure 7: Distribution of the optimalα values obtained from the DSLE-minG0W0 approach forthe GW100 test set with the Tier 4+ basis set.

the recently developed DSLE-minGW approach. Based on the comparison with CCSD(T) ref-

erence data, the results presented here quantify the overall accuracy of different flavors ofGW

calculations for molecular compounds of diverse chemical composition. Overall, our scGW calcu-

lations suggest that the effect of vertex correction may become important for compounds charac-

terized by chemical bonds with a pronounced ionic character(as, for instance, halogenides) or by

nitrogen-lone pair orbital types, as these compounds exhibit the largest deviation from CCSD(T).

Conversely, scGW ionization energies lie typically within 0.3 eV from CCSD(T) for covalently

bonded compounds. The comparison between scGW, scGW0, and qsGW further reveals that dif-

ferent forms of self-consistency may influence the ionization energies and its agreement with the

reference data considerably. We have identified underscreening, density changes and the treatment

of the kinetic energy as reasons for the difference in the different self-consistentGW schemes.

Finally, we have shown that the deviation from CCSD(T) may inpart be attributed to the DSLE

and, correspondingly, the DSLE-minimization procedure recently proposed by some of the au-

thors emerges as a promising way to optimize the starting point of G0W0 calculations to improve

the prediction of ionization energies.

26

Figure 8: Mean absolute error of the deviation from CCSD(T) for theG0W0@PBE(α) ionizationenergies of theGW100 benchmark set as a function ofα. The MAE of DSLE-minGW is reportedas a red solid line. We use Tier 4+ basis sets for our DSLE-minG0W0 calculations.

Acknowledgement

We thank Xinguo Ren and Stephan Kümmel for fruitful discussions. This work was supported

by the Academy of Finland through its Centres of Excellence Programme under project num-

bers 251748 and 284621. M.D. acknowledges support by Deutsche Forschungsgemeinschaft

Graduiertenkolleg 1640 and the Bavarian State Ministry of Science, Research, and the Arts for

the Collaborative Research Network Soltech.

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